Universally reversible \(JC^\ast\)-triples and operator spaces (Q2843991)

A \(JC^{\ast}\)-triple is said to be universally reversible if \(\pi(E)\) is reversible in \(B(H)\) for every triple homomorphism \(\pi:E\rightarrow B(H)\). In [\textit{L. J. Bunce} et al., Math. Z. 270, No. 3--4, 961--982 (2012; Zbl 1251.47063)], it was proved that Cartan factors are universally reversible with the exception of Hilbert spaces of dimension at least 3 and spin factors of dimension at least 5. The universal ternary ring of operators (TRO) of a \(JC^{\ast}\)-triple is a pair \(( T^{\ast}(E), \alpha_{E})\) consisting of an injective triple homomorphism NEWLINE\[NEWLINE \alpha_{E}: E\rightarrow T^{\ast}(E),NEWLINE\]NEWLINE where \(T^{\ast}(E)\) is a TRO generated by \(\alpha_{E}(E)\) (as a TRO) and possessing the universal property that, for each triple homomorphism \(\pi:E\rightarrow B(H)\), there is a (unique) TRO homomorphism \(\tilde{\pi}:T^{\ast}(E)\rightarrow B(H)\) with \(\tilde{\pi}\circ\alpha_{E}=\pi\). In this paper, the authors give a characterization of universal reversibility of a \(JC^{\ast}\)-triple \(E\), proving that this condition is not satisfied if and only if \(E\) has a triple homomorphism onto a Hilbert space of dimension at least 3 or a spin factor of dimension at least 5.NEWLINENEWLINENEWLINEAmong the main results, they prove that every \(JW^{\ast}\)-triple with zero type \(I_{\mathrm{finite}}\) part is a universally reversible \(JC^{\ast}\)-triple, and also give a necessary and sufficient condition for ternary rings of operators (TROs) to be \(JC^{\ast}\)-triples. There is a bijective correspondence between the operator space ideals of \(T^{\ast}(E)\) and the \(JC\)-operator space structures of \(E\), where \(E\) is a universal reversible \(JC^{\ast}\)-triple. If \(\pi: T\rightarrow T\) is a surjective linear isometry of a \(W^{\ast}\)-TRO factor \(T\) not linearly isometric to a \(C^{\ast}\)-algebra, then \(\pi\) is a complete isometry.